multijimbo

A long time ago, I wrote about the most beautiful equation in mathematics and the numbers involved, π and e.  I’ve started slowly working through my backblog and realized that I never finished that particular tale, as I promised.

Looking back on it, it strikes me that I’m not sure I was writing to an appropriate audience.  This is part of a larger internal ponderation I’m having, about writing mathematics in general and writing mathematics for an audience of non-mathematicians in particular.

I have come to believe that spending so much of time thinking about mathematics, and teaching mathematics to others, has shaped how I approach many things.  When I engage in discussions about rules and regulations, process and procedures, I like the ground to be firm beneath my feet.  I like to know the definitions of the terms we use in those discussions, and the rules of argument that we use in our deliberations.

But I think that it does sometimes get in the way when I’m trying to write.  I tend to write very slowly, and there are times I think that my slowness of writing stems from this need to have a clear picture, from beginning to end.  This is relatively straightforward in a piece of mathematics, but much less clear in a work of fiction.  Or a blog.

So back to the matter at hand.  In the previous section, I was talking about levels of complication of numbers.  We started with the rational numbers, quotients of integers.  I then defined the algebraic numbers.  These numbers have the technical definition that they are solutions to equations of the form p(x) = 0, where p(x) is a polynomial with integer coefficients.  Another, heuristic, explanation is that the algebraic numbers are those numbers which can (essentially) specified by a finite collection of integers, much in the same way that rational numbers are specified by pairs of integers.

These levels of complication are related to one another.  Every rational number n/m is an algebraic number, where the polynomial is p(x) = mx – n (since p(m/n) = n (m/n) – m = m – m = 0).  In fact, the rational numbers are the simplest algebraic numbers, if we take as our yardstick of simple to be the degree of the relevant polynomial.

But not every algebraic number is a rational number.  The classical example, classical in the sense of Pythagoras and the ancient Greeks and things we have known for a very very long time, is √2 with associated polynomial p(x) = x^2 – 2.

There are lots of directions we can go from here.  One direction is the question, are there any real numbers which are not algebraic.  Another direction is the question, what other flavours of numbers are there.  And you won’t be surprised to find out, there are many.  Mathematicians have been busy for centuries exploring numbers and their properties.

So how do we demonstrate the existence of real numbers that are not algebraic numbers?  There are hard to construct explicitly, such is their nature.  My favorite demonstration takes us on a detour into a different, and difficult, topic, which is the strange nature infinity.

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For the past 2 years and a bit, I have been in a position of university middle management. My role, as Associate Dean (Education and Student Experience), runs the gamut from the low level operational to the high level strategic.  The breadth of that gamut, I have to say, is one of the reasons I love my role.  The variety of it appeals to me.

But I’ve never received any formal training for my role.  That is not uncommon for such roles in academia, and indeed isn’t uncommon for any of the roles we take on in academia.  I have learned things through my own experience, talking to colleagues, observing and reflecting on my conduct and theirs, my decisions and theirs, my mistakes and theirs.  There is a never stated expectation, not entirely unreasonable, that we academics are reasonably smart people and we can pick things up as we go along.  And we do.

I like to believe that I’m good at what I do.  And when new members of academic staff or postgraduate students ask me, as they sometimes do, how I learned to operate in the ways I operate, whether it’s as a teacher (relatively common), a researcher (much more rare) or as a manager and administrator (again, relatively common), I always start with the same first piece of advice.

Take every opportunity to watch people.  Watch the people you believe are effective, and emulate the things they do well.  But also watch the people you believe are less effective, and actively work not to emulate the things they do that you believe are not effective.  I don’t think we spend enough time learning from things that don’t work.

There are some things I’ve learned to watch out for.  I am confident that nothing on the list below is new.  As noted above, one of the things that we tend to undertake as academics is the reinvention of wheels.  We do this by following the same path as underlies our research.  We start from what we feel are basic principles and we work logically from this starting point.  We do our literature reviews, to find out what is known about our topic of current interest, but we question, and we challenge, and we wonder, can I do better by working from basic principles.

Nonetheless, I’ll list a few of them here.  I would be interested to know what you, dear Reader, think of the items on this list and to suggest relevant additions.

THE LEFT HAND RIGHT HAND PROBLEM:  One issue that bedevils all large organizations is internal communication.  Universities are no exception.  A not-uncommon problem is that 2 (or more) parts of the organization are working to solve a problem, each without the other(s) knowing.  This is rarely deliberate, and often can arise from people being proactive and wanting to do better for the organization.

THE TOO MANY COOKS PROBLEM:  Being an communication issue, this one is related to the previous mentioned issue, and indeed sometimes follows from it.  When several groups are working to address an issue of interest, even when there is communication between them, it’s possible that confusion erupts because too many possible solutions are being proposed and discussed in too many different fora.  The path towards the solution to the issue of interest can become confused, with too many possible overlapping solutions.

THE MOMMY DADDY PROBLEM: Perhaps in the modern age, I should retitle this as the parent 1 parent 2 problem, but to my ear it doesn’t scan as nicely.  A child wants to do or wants to have something and goes to ask one parent, can I do this or can I have this.   The parent, being sensible, says no.

The child, unhappy with that answer and unwilling to accept it, then does what seems to them to be reasonable.  They go ask the other parent, without providing the full disclosure that they had already asked the first parent.

This is my least favorite problem on my list.  The first two are communication problems where the solution, the fix, is reasonably clear.  This one, though, this issue stems as much from attitude as from communication.

THE TAIL AND DOG PROBLEM:  The task before us here is to make sure that we understand what drives the actions we take, and how we remain sure that what is actually driving our actions is what we believe should be driving our actions.

Perspicaciousness is one of my favorite words.  And the only way around the tail and dog problem that I know is to keep the end goal in mind.  We can sometimes lose sight of the shining city on the hill when we are trudging through the slough of despond, but when the darkness falls, we need to search for the glow of the city.

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Those of you who know me, know that I carry around a paper journal and that I do a lot of writing long hand.  I’ve always done a lot of writing long hand, but I was thinking recently about the current generation of students.  What follows will, I suspect, prove to be fodder for further discussions and so I’m going to touch on a lot of things, diving into none of them deeply.  I should also note that what comes below, as all the posts in this blog, are my personal views and speculations.

For coursework, the essays and project work that we ask students to do during the semester, we allow them to use word processors or perhaps mathematical typesetting programmes such as LaTeX.  In fact, we tend to go a bit further and we require them, by and large, to produce electronic versions rather than write things out by hand.   The question is, why?

None of the documents I produce for the consumption of others are written out long hand.  This isn’t just because my handwriting tends to be somewhat indecipherable when I’m writing freely.  It’s also because I find, as I suspect many others find, that drafting and editing and redrafting is far far easier when using a word processor or suchlike.  I find a certain appeal to writing things out by hand, but this is a private appeal.

And so, I have come across a question.  Why do we ask students to write out 2 hour examinations long hand?  For a few subjects, such as mathematics, I see an argument, because typesetting mathematical symbols can be very time consuming.  But for many subjects, we are asking students to do something that they would not ordinarily do, and if fact that we as the teachers might not ordinarily do.

There is a significant logistical answer to this question, namely the difficulty in having enough machines which are sufficiently secure to be appropriate for examination conditions.  One issue here is that having a sufficient volume of machines to use during examination time would be difficult to justify, as they wouldn’t be used at other times of the semester or the other times of year.  Letting students use their own machines, even under supervised conditions, would create the opportunity for some students having greater access to materials and materiel than others, depending on what files folk would have on their machines and what access they would have to the internet, and it would be impossible to effectively police this.  

But this then just leads us back to the question that started us along this particular line.  Namely, why do we use 2 hour examinations at the end of the semester to gauge what students have learned over the course of a semester.  This is a much more difficult question.  On the one hand, having an end of semester examination allows us as teachers to test students under controlled conditions: do these questions, in this time, starting either with a blank examination book or having access to some set collection of material, formula sheets, et cetera.

But on the other hand, this sort of timed examination is not the sort of thing that most people face once they finish their university programme.   A few people will, when they take professional examinations, but a relatively few.  And so I’m starting to speculate, why do we need end of semester examinations, and if we move to something else, what might that something else be. 

I am not the first person to think about this, not by a long way, and the increase of coursework only or coursework majority modules is a significant sign that this is not only something that people are thinking about, but it is something that people are working on.  And so let’s leave this one for the moment.

The next question, moving back one more step, is why we structure the information we deliver by the semester.  Again, there are strong reasons for doing so, and not just the inertia of history keeping us doing the same thing over time.  But I do think that there is an interesting point here for ponderation, namely how we structure the ways in which we structure the material we deliver.  And this isn’t even getting into the question of how we deliver material, which is a massive question in its own right.

One thing that is interesting is that the deeper we dig into these questions and the more we start to question some of the fundamental ways in which higher education institutions are structured educationally, there are institutions doing things differently.  But I do think that universities with large numbers of students face particular challenges.  Over time, we’ll try and come back to some of these, but I think I need to do a bit of reading first, to find out what the current state of the art is.

I suspect that for you, as for me, this has been a somewhat unsatisfactory point at which to conclude.  I have raised a lot of questions, with no clear answers, but I do think that the shape of higher education is changing.  And I think it’s going to be an exciting ride.

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So where to start?  After seeing Star Wars episode VII: The Force Awakens, for what will be the first of many times, I was struck by an idea for something to blog about.  Start with the joke about the similarities between the Force and duct tape, replace the duct tape with mathematics, and thusly entertain the readers.

But then, procrastination (and see something I’ve been meaning to do for a while for more on that particular topic) and the idea got put on the pile of things awaiting attention.  The more I thought about it, the more I began to doubt my original plan.  Comparing duct tape and mathematics is only one part of the issue.  The other is that I am beginning to doubt that the Force has a light side and a dark side.

Heretical as this might sound, at least to those of you who declared yourselves as Jedi in the previous census, I don’t believe that the Force has a light side and a dark side.

Before going further, though, I should say that I have not done any reading through the now extensive literature of the extended Star Wars universe, nor watched any of the animated films or series.  At least, none beyond The Splinter of the Mind’s Eye, which came out before episode V.   So perhaps this is something that’s been kicked around therein.

So why no light side and no dark side.  First of all, I believe that nature abhors such a dichotomy.  A light side and a dark side leaves no space for the ambiguity that pervades our intentions and actions.

But even beyond this, I am not convinced that we are so significant to the Force that it moulds itself to our petty concerns and squabbles over power.  I prefer to think that what we view as the light side and the dark side are the reflections of ourselves that we see in the mirror of the Force.  We create the light side and we create the dark side, and the Force just abides.

I will admit that part of the reason I prefer this interpretation is that this makes our relationship to the Force into an active relationship.  We are not passive tools of some dichotomy that exists beyond us.  Instead, we make a choice of the direction towards which we tilt.  It also allows for the possibility of a dynamically unstable middle ground, a grey servant of the Force.

The necessity of the Sith then is that we each have something of a dark side, the thoughts and desires that inform the entertainments with which we like to spend our time.  The necessity of the Sith is that we are human, or one of the other remarkably similar races that fill the ranks of both the Jedi and the Sith.

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